Investigation of Azimuthal Asymmetries in Charged and ... · In this thesis the anisotropic transverse ﬂow v 2 of charged and strange particle species ... 6 FLOW ANALYSIS OF CHARGED

Investigation of Azimuthal Asymmetries

in Charged and Strange Particle Distributions

from CERES

Jovan Miloševíc

Physikalisches Institut der Universität Heidelberg

2005

Dissertationsubmitted to the

Combined Faculties for the Natural Sciences and for Mathematicsof the Ruperto-Carola University of Heidelberg, Germany

for the degree ofDoctor of Natural Sciences

presented by

MSc in physics Jovan Milǒsevíc

born in Vr šac, Serbia

Oral Examination: 14.02.2006

Investigation of Azimuthal Asymmetries

in Charged and Strange Particle Distributions

from CERES

Referees: Prof. Dr. Johanna Stachel

Prof. Dr. Norbert Herrmann

Diese Doktorarbeit stellt die Messung des anisotropen transversalen Flussesv2 gelad-ener und seltsamer Teilchen mit dem CERES Spektrometer vor.Gezeigt werden dieelliptischen Fluss Messungen der Teilchen�, K0S, �� und p in Pb-Au Kollisionen beihöchsten SPS Energien. Mit einem Pseudorapiditätsbereich von� = 2:05 � 2:70 beivoller azimuthaler Akzeptanz und mit einerpT Sensitivität von bis zu 4 GeV/c können mitden CERES Daten hydrodynamische Modelle getestet werden.v2 wird als Funktion derZentralität, Rapidität, Pseudorapidität und des Transversalimpulses für die verschiedenenTeilchensorten diskutiert. Die Messungen werden mit Ergebnissen des NA49 Experi-mentes und mit hydrodynamischen Rechnungen verglichen. Ferner werden Vergleiche zuden STAR und RHICH Beobachtungen angestellt. Bei kleinempT wird der Massenord-nungseffektv2(�) < v2(K0S) < v2(��) beobachtet. Ein entgegengesetztes Verhalten zeigtsich bei hohempT . Um ein tieferes Verständniss für die Ursachen des Skalenverhal-tens des kollektiven Flusses mit der Anzahl an Konstituentenquarks und mit der transver-salen Rapidität zu erhalten, wird die von der HydrodynamikvorhergesagteyfsT -Skalierungdurchgeführt. Vergleiche differentieller Flussmessungen verschiedenster Teilchen mitverschiedenen Szenarien an Skalenverhalten ermöglicht eine Aussage über die Ursachendes Flusses, sowie über die frühesten Stadien der Kollision.

In this thesis the anisotropic transverse flowv2 of charged and strange particle speciesmeasured by the CERES experiment is investigated. The�, K0S, �� and proton ellipticflow measurements from Pb+Au collisions at the highest SPS energy are presented. Thedata, collected by the CERES experiment which covers� = 2:05 � 2:70 with full 2�azimuthal acceptance andpT sensitivity up to 4 GeV/c, is used to test hydrodynamicalmodels. The value ofv2 as a function of centrality, rapidity, pseudorapidity andpT ispresented for different particle species. The obtained measurements are compared withresults from the NA49 experiment and with hydrodynamical calculations. Also the resultsare compared withv2 values observed with STAR at RHIC. The mass ordering effect wasobserved:v2(�) < v2(K0S) < v2(��) at smallpT , while at highpT it is opposite. Inorder to get better insight into the origin of the collectiveflow scaling to the number ofthe constituent quarks and the transverse rapidityyfsT scaling predicted by hydrodynamicswere performed. Testing the differential flow measurementsof different particle speciesagainst different scaling scenarios may yield additional information about the origin offlow as well as about the early stage of the collision.

Acknowledgments

To my son and wife, to my parents, to my brother and his family for their love andsupport that keeps me going.

To all my professors at the Physikalisches Institut in Heidelberg, at the Institute ofPhysics and the Faculty of Physics in Belgrade; thank you forteaching me so much andfor making me realize there is so much more to learn.

I am grateful to Ana Marı́n who was willing to read my thesis and to give usefulremarks in order to make it better. Also I would like to thank to Wilrid Ludolphs whoprovided me a code for the secondary vertex reconstruction which was used in theK0Selliptic flow analysis.

And thank to my friends here at the Physikalisches Institut and at the Gesellschaft fürSchwerionenforschung who were helping me to finish this thesis.

Contents

1 INTRODUCTION 11.1 Quark-Gluon Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Experimental Search for the QGP . . . . . . . . . . . . . . . . . . . . .11.3 Overview of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 QGP AND HEAVY-ION COLLISIONS 52.1 The Lattice Quantum Chromo Dynamical Predictions . . . . .. . . . . . 52.2 Geometry and Space-Time Evolution of a Heavy-Ion Collision . . . . . . 62.3 Heavy-Ion Collisions and Signatures of the QGP . . . . . . . .. . . . . 92.4 Collective Flow as Signature of the QGP . . . . . . . . . . . . . . .. . . 10

3 METHODS IN FLOW ANALYSIS 133.1 Sphericity Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Mean Transverse Momentum in the Reaction Plane . . . . . . . .. . . . 14

3.2.1 The Reconstruction of the Reaction Plane . . . . . . . . . . .. . 153.2.2 Flattening of thedN=d� Distribution . . . . . . . . . . . . . . . 173.2.3 The Reaction Plane Resolution . . . . . . . . . . . . . . . . . . . 18

3.3 Two-particle Correlations . . . . . . . . . . . . . . . . . . . . . . . .. . 183.4 Fourier Analysis of the Azimuthal Distributions . . . . . .. . . . . . . . 193.5 The Cumulants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.5.1 Integrated Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.5.2 Differential Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.6 The Lee-Yang Zeroes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.6.1 Integrated Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.6.2 Differential Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.7 The Method Used in this Analysis . . . . . . . . . . . . . . . . . . . . .30

4 EXPERIMENTAL SETUP AND DATA USED 314.1 The CERES Experimental Setup . . . . . . . . . . . . . . . . . . . . . . 31

4.1.1 The Target and the Beam/Trigger Detectors . . . . . . . . . .. . 324.1.2 The Silicon Drift Detectors . . . . . . . . . . . . . . . . . . . . . 324.1.3 The RICH Detectors . . . . . . . . . . . . . . . . . . . . . . . . 344.1.4 The Time Projection Chamber . . . . . . . . . . . . . . . . . . . 34

4.2 The Calibration and Production of Data . . . . . . . . . . . . . . .. . . 354.2.1 The Ballistic Deficit Correction . . . . . . . . . . . . . . . . . .36

xi

xii CONTENTS

4.2.2 The SDD Efficiency . . . . . . . . . . . . . . . . . . . . . . . . 374.3 Data Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3.1 Particle Identification . . . . . . . . . . . . . . . . . . . . . . . . 414.3.2 ThedN=d�, dN=dpT anddN=d� distributions . . . . . . . . . . 424.3.3 The Momentum Resolution . . . . . . . . . . . . . . . . . . . . 43

4.4 Centrality Determination . . . . . . . . . . . . . . . . . . . . . . . . .. 45

5 FLOW ANALYSIS OF SIMULATED DATA 495.1 Flowmaker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.2 The Data Simulated by the Flowmaker . . . . . . . . . . . . . . . . . .. 495.3 Flow Analysis of the Simulated Data Using the Reaction Plane Method . 515.4 Cumulant Analysis of the Simulated Data . . . . . . . . . . . . . .. . . 535.5 Lee-Yang Zeroes Analysis of the Simulated Data . . . . . . . .. . . . . 55

6 FLOW ANALYSIS OF CHARGED PARTICLES 596.1 Particle Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596.2 Determination of the Reaction Plane . . . . . . . . . . . . . . . . .. . . 606.3 Elliptic Flow of Pions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646.4 Elliptic Flow of Identified Protons . . . . . . . . . . . . . . . . . .. . . 696.5 HBT Effects on the�� Elliptic Flow Measurements . . . . . . . . . . . . 69

7 FLOW ANALYSIS OF � PARTICLES 777.1 Particle Selection and� Reconstruction . . . . . . . . . . . . . . . . . . 777.2 Reaction Plane Determination and its Resolution . . . . . .. . . . . . . 847.3 Elliptic Flow of� Particles . . . . . . . . . . . . . . . . . . . . . . . . . 87

8 FLOW ANALYSIS OF K0S PARTICLES 918.1 Particle Selection andK0S Reconstruction . . . . . . . . . . . . . . . . . 918.2 Reaction Plane Determination and its Resolution . . . . . .. . . . . . . 958.3 Elliptic Flow ofK0S Particles . . . . . . . . . . . . . . . . . . . . . . . . 96

9 COMPARISONS AND SCALINGS 999.1 Comparison with Hydrodynamical Model . . . . . . . . . . . . . . .. . 999.2 Comparison with STAR and NA49 experiment . . . . . . . . . . . . .. 1019.3 Mass Ordering Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1029.4 Scaling to the Number of Constituent Quarks . . . . . . . . . . .. . . . 1039.5 Scaling to the Flavor Transverse RapidityyfsT . . . . . . . . . . . . . . . 104

10 CONCLUSIONS 105

A VARIABLES 107A.1 Rapidity and Pseudorapidity . . . . . . . . . . . . . . . . . . . . . . .. 107

B FINITE GRANULARITY IN dN�(K0S)=d� 109B.1 Correction for the Finite Granularity indN�(K0S)=d� distributions . . . . . 109

List of Figures

2.1 The normalized energy density�=T 4 and pressurep=T 4 vs temperature obtained fromLQCD for 0, 2 and3 light quark flavors, as well as for2 light +1 heavier (strange)quark flavors. Horizontal arrows on the right show the corresponding values for Stefan-

Boltzmann gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 The LQCD results for non-zero chemical potential [1] suggest the existence of a crit-

ical point well above RHIC chemical potential values. The solid line represent a 1-st

order phase transition, while the dotted one indicates a crossover transition between two

phases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 A cartoon presentation of a central (a) and a peripheral (b) collision inx� z plane of the

collision. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Schematic space-time evolution of a central heavy-ion collision with a QGP phase formed

during the collision. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 The QCD phase diagram of the hadronic matter [2]. The points show chemical freeze-

out of hadrons extracted from different heavy-ion experiments. . . . . . . . . . . . 102.6 Hydrodynamical predictions ofv2 excitation functions (left axis) and radial flow velocityhhv?ii (right axis) for non-central Pb+Pb collisions [3]. . . . . . . . . . . . . . . 112.7 The beam energy dependence of the elliptic flow. The figure is taken from [4]. . . . . 12

3.1 A schematic view of a collision of two nuclei in the transverse plane. The spatial asym-metry, showed at the top, is transformed into a momentum asymmetry, showed at the

bottom, due to the pressure gradient which was built up during the collision. . . . . . 14

4.1 The CERES/NA45 experimental setup in year 2000 the data taking period. . . . . . . 324.2 The Silicon Drift Detectors operate on base of electrons drifting in a radially symmetric

electric field towards the edge of the detectors.. . . . . . . . . . . . . . . . . . 334.3 An interlaced structure of the anode divided into 5 pieces.. . . . . . . . . . . . . 334.4 The cylindrical Time Projection Chamber operates in a radial drift field. . . . . . . . 354.5 The ballistic deficit before (top) and after (bottom) the correction in the case of SDD1

(left) and SDD2 (right). Data are fitted with a second order polynom. . . . . . . . . 364.6 Top: The signal (S) and normalized combinatorial background (Bnorm) of matched

SDD tracks to the referent TPC track in� (left) and� (right) direction. Middle: The ratiobetween the signal and normalized combinatorial background. Bottom: The difference

between the signal and normalized combinatorial background. . . . . . . . . . . . 384.7 The SDD efficiencyvscentrality expressed in the TPC multiplicity.. . . . . . . . . 394.8 The efficiency�SDD vspolar (�) and laboratory azimuthal angle (�). . . . . . . . . . 404.9 The efficiency�SDD vsmomentump. . . . . . . . . . . . . . . . . . . . . . . 40

xiii

xiv LIST OF FIGURES

4.10 The momentum-dE=dx particle distribution for all detected charged particles.Full linesrepresent a nominal energy loss calculated by using the Bethe-Bloch formula. Within

dashed line (which corresponds to�1:5� confidence) are chosen�+. The same is in thecase of��. Even more,� particles as well as low momentum protons and deuterons areclearly separated by theirdE=dx. . . . . . . . . . . . . . . . . . . . . . . . . 42

4.11 The pseudorapidity (left) andpT (right) distribution of particles detected in the TPC.TPC and SDD track segments are matched within a3� window. . . . . . . . . . . . 43

4.12 The distribution of the laboratory azimuthal angle�lab in the TPC. . . . . . . . . . 434.13 The momentum resolution obtained by using a Monte Carlo simulation of the detector.

The Figure was taken from [5]. . . . . . . . . . . . . . . . . . . . . . . . . . 444.14 The TPC multiplicity distribution for all events used in theelliptic flow analysis. The

distributions, obtained with different trigger conditions, are normalized to the minimum

bias distribution in the high TPC multiplicity region.. . . . . . . . . . . . . . . . 454.15 Left: the correlation between the TPC and SDD multiplicity.Right: The Gaussian mean

value of the projection to the SDD multiplicity axisvs the TPC multiplicity. A linear fit

describes the obtained correlation.. . . . . . . . . . . . . . . . . . . . . . . . 454.16 The correspondence between the SDD multiplicity and the geometrical cross section�=�geo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.17 The TPC multiplicity distribution is divided into two centrality bins in which the elliptic

flow analysis is performed. They are characterized with the weighted mean centralityh�=�geoi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.1 Multiplicity distribution from the Flowmaker simulated data. . . . . . . . . . . . . 505.2 dN=d� (left) anddN=dpT (right) distribution from the Flowmaker simulated data.. . 505.3 dN=d� distribution from the Flowmaker simulated data.. . . . . . . . . . . . . . 515.4 The true (open circles) and reconstructed (closed circles)second Fourier coefficientvs�. 525.5 The true (open circles) and reconstructed (closed circles)second Fourier coefficientvspT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.6 The integratedv1 (left) andv2 (right) dependence on multiplicity obtained from the first

three cumulants.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.7 Top: the absolute values of the generating functionsGn in the case of the directed (left)

and elliptic (right) flow plotted against ther value. Bottom: the same as at the top butzoomed at the position of the minimum.. . . . . . . . . . . . . . . . . . . . . 55

5.8 The reconstructed value ofv2 vs� (top) andvspT (bottom) for different� values (left)and averaged over different� values (right). . . . . . . . . . . . . . . . . . . . . 56

6.1 The momentum-dE=dx particle distribution for the selected pions, protons and deuterons.Full lines represent a nominal energy loss calculated by theBethe-Bloch formula. Within

dashed lines (which correspond to�1:5� confidence) are chosen�+. The same is in thecase of��. The low momentum protons and deuterons are clearly separated by theirdE=dx. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.2 The distribution of slices in� space. The size of a slice is3:6Æ. Each fourth of themforms a group denoted with 1, 2, 3 or 4. . . . . . . . . . . . . . . . . . . . . . 60

6.3 Raw reaction plane distribution calculated fromXn andYn coefficients forn = 1 (top)andn = 2 (bottom) in the second (left) and fourth (right) slice.. . . . . . . . . . . 61

LIST OF FIGURES xv

6.4 The reaction plane distribution after applying the shifting method forn = 1 (top) andn = 2 (bottom) in the second (left) and fourth (right) slice.. . . . . . . . . . . . . 616.5 The reaction plane distribution after applying the shifting and the Fourier method of

flattening forn = 1 (top) andn = 2 (bottom) in the second (left) and fourth (right) slice.626.6 Shifting coefficientsXn andYn for n = 2 in the first two centrality bins (up and down)

for the whole event (left), and subeventsa (center) andb (right) versus the unit number. 626.7 Reaction plane resolution in case of the second harmonic forall centrality binsvsunit

number. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.8 The correction factorsvscentrality expressed via TPC multiplicity for the 2 subeventsand the ’slice’ method. Due to the roughly double multiplicity, correction factors in the

2 subevents method are� p2 times smaller then in case of the ’slice’ method.. . . . 646.9 Thev2 valuesvs� (left) andy (right) for pions from all centralities taken together. The

results are obtained using the 2 subevents method.. . . . . . . . . . . . . . . . . 65

6.10 The pionv2(pT ) from all centralities taken together. The results are obtained using the2 subevents method and not corrected for the HBT effect.. . . . . . . . . . . . . . 65

6.11 Thev2 valuesvscentrality. The closed circles denote the present analysis, while the opencircle represents the older (completely independent) analysis. The result is not corrected

for the HBT effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.12 The pion elliptic flowvs rapidity (left) andpT (right) for central (closed circles) andsemicentral (open circles) collisions.. . . . . . . . . . . . . . . . . . . . . . . 66

6.13 The pion elliptic flowvs pseudorapidity (left) and rapidity (right) calculated using the’slice’ (closed circles) and subevent method (open circles). . . . . . . . . . . . . . 67

6.14 The pion elliptic flowvstransverse momentum calculated using the ’slice’ and 2 subeventsmethod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.15 The ratio (left) and the difference (right) betweenv2(pT ) calculated using the ’slice’method and the 2 subevents method.. . . . . . . . . . . . . . . . . . . . . . . 68

6.16 The�+ (closed circles) and�� (open circles) elliptic flowvspseudorapidity (left) andrapidity (right) for all centralities taken together.. . . . . . . . . . . . . . . . . . 68

6.17 The identified proton elliptic flowvspT . . . . . . . . . . . . . . . . . . . . . . 696.18 Left: the correlation coefficientHBT2 vspT calculated via Eq. (6.13). Right: the appar-

entvHBT2 (pT ) pion elliptic flow arising only from the HBT correlations calculated fromHBT2 (pT ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726.19 The pion elliptic flowvspT before (closed circles) and after (open circles) correction for

the HBT effect. A parabolic (p2T ) fit is indicated with a full line. . . . . . . . . . . . 736.20 The integrated pion elliptic flowvs �=�geo calculated using the 2 subevents method

corrected for the HBT effect. . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.21 The ratio between the integrated pion elliptic flow uncorrected and corrected for theHBT effectvs�=�geo calculated using the 2 subevents method.. . . . . . . . . . . 74

6.22 The pion elliptic flowvs the transverse momentum calculated using the 2 subeventsmethod for 3 different centralities before (closed circles) and after (open circles) correc-

tion for the HBT effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

xvi LIST OF FIGURES

7.1 Left: Partially identified�� and protons in the case of a sharp cut on the transversemomenta of positive particles and a sharp opening angle cut (Run I). Right: Partially

identified�� and protons in the case were combinedpT dependent opening angle�p��cuts have been applied.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

7.2 Armenteros-Podolanski plot shows�, �� andK0S reconstructed from the experimentaldata. The Figure is taken from [6].. . . . . . . . . . . . . . . . . . . . . . . . 79

7.3 Left: Run I. Top: the invariant mass distribution of the signal and the normalized com-binatorial background. In the region of the� mass a pronounced signal is observed.Bottom: the invariant mass distribution of the signal aftersubtraction of the normalized

combinatorial background. Right: Run II. Top: A small enhancement of the signal is

visible in the region of the� mass. Bottom: the invariant mass distribution of the signalleft after subtraction of the normalized combinatorial background. . . . . . . . . . . 80

7.4 Left: mass of� in function ofpT for different rapidities displayed with different sym-bols. Right: the same dependences in the case of width of� . . . . . . . . . . . . 81

7.5 Left: S=B as a function of applied cuts (for the correspondence between the depictedpoints and the applied cuts see the text below). Right: the same dependences in the case

of the significanceS=pB. . . . . . . . . . . . . . . . . . . . . . . . . . . . 827.6 Left: S=B as a function ofpT for different rapidities displayed with different symbols.

Right: the same dependences in the case of the significanceS=pB. . . . . . . . . . 827.7 Distribution of accepted� in y � pT space. . . . . . . . . . . . . . . . . . . . . 837.8 Top: � reconstructed for1:62 � y � 1:69, 0:675 � pT � 0:8 GeV/c and15Æ �� 30Æ. Bottom: Elliptic flow pattern reconstructed from the� yield in � bins forpT � 2:7 GeV/c. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837.9 Example of flattening of the calculated reaction plane(�) in one of 6 centrality bins

(Run II). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857.10 CorrecteddN=d� distribution for 9 different centralities (Run I) as a function of the

reaction plane angle� calculated using the Eq. (7.2). . . . . . . . . . . . . . . . 857.11 Correction factors, as the inverse values of the reaction plane resolutions calculated using

the Eq. (3.13) are presented as a function of centrality.. . . . . . . . . . . . . . . 867.12 � elliptic flow vspT for all centralities taken together.. . . . . . . . . . . . . . . 877.13 � elliptic flow vspT for the semicentral (left) and central (right) collisions.. . . . . . 877.14 � elliptic flow ıvsy in the case of semicentral events.. . . . . . . . . . . . . . . 887.15 � elliptic flow vscentrality. . . . . . . . . . . . . . . . . . . . . . . . . . . . 887.16 � elliptic flow vspT from Run I and Run II calculations.. . . . . . . . . . . . . . 898.1 Left: The invariant mass distribution of the signal (red line) and the normalized com-

binatorial background (black line). Right: The invariant mass distribution of the signal

after subtraction of the normalized combinatorial background. . . . . . . . . . . . 938.2 Left: mass ofK0S as function ofpT for different rapidities displayed with different

symbols. Right: the same dependence in the case of the width of K0S . . . . . . . . . 938.3 Distribution of acceptedK0S in y � pT space. . . . . . . . . . . . . . . . . . . . 948.4 Top: K0S reconstructed for1:62 � y � 1:69, 0:675 � pT � 0:8 GeV/c and15Æ �� 30Æ. Bottom: Elliptic flow pattern reconstructed from the� yield in � bins forpT � 2:1 GeV/c. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

LIST OF FIGURES xvii

8.5 Correction factors in theK0S elliptic flow analysis using the Eq. (3.13) are shown as afunction of centrality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

8.6 K0S elliptic flow vspT for all centralities taken together. . . . . . . . . . . . . . . 968.7 K0S elliptic flow vspT for the semicentral (left) and central (right) collisions.. . . . . 968.8 K0S elliptic flow vsy in the case of semicentral events.. . . . . . . . . . . . . . . 979.1 Comparison between the hydrodynamical calculation and theCERES experimental re-

sults on� elliptic flow in semicentral (left) and central (right) events. . . . . . . . . 1009.2 Comparison of� elliptic flow measured by CERES, STAR and NA49.. . . . . . . . 1019.3 Comparison ofK0S elliptic flow measured by CERES and STAR.. . . . . . . . . . 1019.4 Comparison between the elliptic flow magnitude of the��, low momentum protons,�,

andK0S emitted in semicentral events.. . . . . . . . . . . . . . . . . . . . . . 1029.5 Comparison between elliptic flow magnitude scaled to the number of the constituent

quarks for the��, low momentum protons,�, andK0S emitted in semicentral events.. 1039.6 Comparison between elliptic flow magnitude scaled to the transverse rapidityyfsT for the��, �, protons andK0S emitted in semicentral events.. . . . . . . . . . . . . . . 104

xviii LIST OF FIGURES

List of Tables

4.1 The Gaussian width value of the pure signal distributionin � and� direc-tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

6.1 The input values for�, Rs, Ro andRL for the HBT correction of theintegratedv2. The input values are obtained by averaging over centralitieswith �=�geo � 15% and overkT � 0:6 GeV/c. . . . . . . . . . . . . . . 72

6.2 The input values for�, Rs, Ro andRL for the HBT correction of theintegrated elliptic flow. The input values are obtained by averaging overkT � 0:6 GeV/c in different centrality bins. . . . . . . . . . . . . . . . . 74

9.1 The mass ordering effect betweenv2 of �, K0S, and�� at the top SPSenergy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

9.2 Thekm values for��,K0S and� particle. . . . . . . . . . . . . . . . . . 104

xix

xx LIST OF TABLES

Chapter 1

INTRODUCTION

1.1 Quark-Gluon Plasma

The experimental study of nuclear matter exposed to extremely high temperatures anddensities offers a unique opportunity for obtaining information concerning strongly in-teracting many-body systems. Most important is the searching for the predicted phasetransition to Quark-Gluon-Plasma (QGP) [7]. QGP is defined as a (locally) thermallyequilibrated state of matter in which quarks and gluons are deconfined from hadrons, sothat color degrees of freedom become manifest over nuclear,rather than merely nucle-onic, volumes. In nature, a transition from QGP to hadronic matter has probably under-gone10�6 � 10�5 s after Big Bang [8, 9]. Now, the above mentioned systems probablyexist in astrophysical objects, like neutron stars and collapsing supernovae [10–12]. Inthat new state of matter the chiral symmetry is restored. From all written above it is clearthat the study of the QGP is of common interest to particle andnuclear physics, as wellas for astrophysics and cosmology.

1.2 Experimental Search for the QGP

In the laboratory, strongly interacting many-body systemsat extreme conditions can beproduced and investigated using heavy-ion collisions at high colliding energies. Due tothat such collisions are investigated already three decades. To study in a systematic waysystems created in such collisions many experiments were designed and built. It startedwith accelerating of relatively light projectiles as He, C,Ne up to Ar at BEVALACin Berkeley, the USA, and in the USSR started to work the LHE synchrophasotron inthe Laboratory for High Energies at the Joint Institute for Nuclear Research (JINR) inDubna. These experiments continued in the eighties with accelerating of heavier nucleiwith the SchwerIonen-Synchrotron (SIS) at the Gesellschaft für Schwerionenforschung(GSI) in Darmstadt (Germany) and the Alternating Gradient Synchrotron (AGS) in theBrookhaven National Laboratory (BNL) in the USA. At CERN (the acronym ofCon-seil Euroṕeen pour la Recherche Nucléaire) the Super Proton Synchrotron (SPS) wasbuilt. They were operating with an incident energy from several hundreds MeV/c (at BE-VALAC) up to 200 GeV/c per nucleon (at SPS). Since that time many experiments used

1

2 CHAPTER 1. INTRODUCTION

these facilities in investigations of the above mentioned physical systems. At incidentenergies between 1 and 200 GeV/c temperatures between 50 and160 MeV are achievedand baryon densities up to 10 times higher than in the case of ’normal’ nuclear matter(�0 = 0:167fm�3) [13]. The needness for even higher colliding energies led to designingand building of a new accelerator, Relativistic Heavy Ion Colider (RHIC) in BNL whichstarted to operate in the year 2000. Currently, under construction, is a new, most powerfulLarge Hadron Collider (LHC) at CERN.

The basic aim of current and future experiments with heavy-ion collisions is searchingfor signatures of the phase transition between the QGP and the hadronic matter and forthe QGP itself. The information about the QGP formation in the early stage of the colli-sion is carried by electromagnetic (thermal photons and dileptons) and hadronic (enlargedproduction of strangeness, suppressedJ= production and highpT jets) signatures. Thenext, but not less important task, is to answer the question which Equation of State (EoS)governs the behavior of matter in the QGP phase. Due to that, the investigation of col-lective effects in nucleus-nucleus collisions takes an important role. The high (spatiallyanisotropic) pressure created during the non-central collision results in later fast expan-sion of matter created in such a collision and in appearance of a collective flow.

Although different types of collective flow, as longitudinal, radial and anisotropictransversal flow are investigated separately, they are in fact interconnected and representdifferent manifestations of the same phenomenon - the collective expansion of matter cre-ated in the nucleus-nucleus collision. The longitudinal flow is an ordered expansion of thesystem along the beam axis and analysis of rapidity distributions can show its existence.The radial flow is introduced in order to explain the distributionsvs transverse massmTfor different kinds of particles. The investigation of the radial flow can give the infor-mation about the temperature achieved in the system. The anisotropic transverse flowappears as a dominant emission of particles in a certain direction in the transverse planeof the collision.

One of the experiments which has been used in one of the facilities mentioned at thebeginning of this Section is the ChErenkov Ring Electron Spectrometer (CERES) de-signed to measure low-masse+e� pairs created in proton-nucleus and nucleus-nucleuscollisions at SPS [14–16]. Dileptons have a particular significance as a probe for hot anddense matter due to the fact that they, in contrast to the hadrons, interact only electromag-netically. Hence, they probe the early stage of the collision.

Beside the dilepton signals, the CERES experiment is also able to detect chargedhadrons and to measure their momenta with high precision. This allowed to perform aninvestigation of another probe of the early stages of the heavy ion collisions - so calledanisotropic transverse flow and especially, elliptic flow. The investigation of the ellipticflow of �, K0S, protons and pions will be the main aim of the analysis presented in thisthesis. The elliptic flow itself appears as an azimuthally anisotropical emission of particleswith respect to the reaction plane of the collision. This is acollective effect created due toan anisotropic pressure gradients built up as a consequenceof a geometrically anisotropicshape of the overlapping zone of the colliding nuclei. The information which the ellipticflow can provide could be used to get some insight about the EoSof the nuclear matterunder study [17].

1.3. OVERVIEW OF THIS THESIS 3

1.3 Overview of this Thesis

In this thesis the analysis of the anisotropic transverse flow of charged (�� and protons)and strange (� andK0S) particles emitted in Pb+Au collisions at the highest SPS energy(158 AGeV/c in the laboratory system) was performed.

This thesis is organized in the following way. In Chapter 2 are described the basicfeatures of QGP as well as the signatures of the QGP. The special emphasis is set onthe elliptic flow as a signature for the QGP. Chapter 3 gives anoverview of methodsdeveloped and used in the anisotropic flow analysis startingfrom the oldest, and nowan obsolete one,sphericity tensormethod up to the newest one, the so called method ofLee-Yang zeroes. Chapter 4 describes the CERES experimental setup. In Chapter 5 ispresented theFlowmaker, a Monte Carlo flow simulator, together with the results fromthe simulated data. The results from the analysis of the experimental data are presentedin Chapters 6 - 8. In these, main Chapters of this thesis is described the analysis of theanisotropic transverse flow of charged (�� and protons) and strange (� andK0S) particles.The obtained results are also compared to the hydrodynamical model and to the resultson the flow analysis from the other experiments. These comparisons are the contents ofChapter 9. Chapter 9 also contains the results on the scalingproperties of the elliptic flowobtained from the CERES data. Finally, Chapter 10 contains the conclusions and outlook.

4 CHAPTER 1. INTRODUCTION

Chapter 2

QGP AND HEAVY-ION COLLISIONS

2.1 The Lattice Quantum Chromo Dynamical Predictions

The phase diagram of strongly interacting bulk matter in theregime of high energy densityand temperature should be described by Quantum Chromo Dynamics (QCD). In a simplepicture of non-interacting massless quarks and gluons, theStefan-Boltzmann (SB) pres-sure (pSB) at zero chemical potential is given by the number of degreesof freedom [18]:pSBT 4 = [2(N2 � 1) + 72NNf ℄�290 (2.1)whereN is the number of colors andNf is the number of quark flavors. Refinementto this basic picture incorporates color interactions on the partonic level, non-vanishingquark masses and finite chemical potential. In order to solvethe problem, the QCD cal-culations are done on a spacetime lattice (LQCD). In order toextract predictions, theLQCD result had to be extrapolated to the continuum (latticespacing! 0), chiral (actualcurrent quark masses) and thermodynamic (large volumes) limits. The LQCD investiga-tions [19] show that matter with zero baryon density undergoes a phase transition at thecritical temperature ofT = (173� 15) MeV from a color-confined hadron resonance gas(HG) to a color-deconfined QGP. The critical energy density� = 0:7 GeV=fm3 roughlycorresponds to the energy density in the center of a proton. Fig. 2.1 shows the normal-ized energy density�=T 4 and pressurep=T 4 vs temperature obtained from the LQCD for0, 2, 3 light and1 heavier (strange) quark flavor [19]. One can see that when thephasetransition occurs the normalized energy density grows dramatically by roughly one orderof magnitude over a rather narrow temperature interval, while the normalized pressure iscontinuous and grows more gradually. Both saturate at about75 � 85% of the Stefan-Boltzmann value for an ideal gas of non-interacting quarks and gluons. The energy den-sity reaches the saturation value very quickly (at about1:2 T) while the pressure growsslower and reaches the saturation at higher temperatures. The Lattice calculations showthat for temperatures above2 T, the EoS corresponds to the equation of state of an idealgas of massless particles, i.e.� = 3p. For temperatures below2 T, the deviation from theSB limit indicates substantial remaining interactions among the quarks and gluons in theQGP phase.

5

6 CHAPTER 2. QGP AND HEAVY-ION COLLISIONS

0

2

4

6

8

10

12

14

16

100 200 300 400 500 600

T [MeV]

ε/T4εSB/T

4

Tc = (173 +/- 15) MeV εc ~ 0.7 GeV/fm

3

RHIC

LHC

SPS 3 flavour2 flavour

‘‘2+1-flavour’’

0

1

2

3

4

5

100 200 300 400 500 600

T [MeV]

p/T4 pSB/T4

3 flavour2+1 flavour

2 flavourpure gauge

Figure 2.1: The normalized energy density�=T 4 and pressurep=T 4 vstemperature obtained from LQCDfor 0, 2 and3 light quark flavors, as well as for2 light +1 heavier (strange) quark flavors. Horizontal arrowson the right show the corresponding values for Stefan-Boltzmann gas.

.

Beside the predictions described above, here below are listed the other LQCD predic-tions

1. Above the critical temperature the effective potential between a heavy quark-anti-quark pair is a screened Coulomb potential with screening mass which rises with thetemperature [20]. That is not in accordance with perturbative QCD expectations.The increasing of the screening mass leads to a shortening ofthe range of theq�qinteraction and to suppression of the charmonium production [21,22].

2. The phase transition is also accompanied by a chiral symmetry restoration [23]. Thereduction in the chiral condensate leads to variations in in-medium meson masses.

3. The kind of the phase transition strongly depends on the number of the dynamicalquark flavors included in the calculation and on the quark masses [24]. The realisticcalculations with two light quarks (u andd) and one heavier (s) at zero chemicalpotential gives a crossover type of transition without discontinuities in thermody-namical observables.

4. The calculations at non-zero chemical potential suggestthe existence of a criticalpoint such as illustrated in Fig. 2.2 [1]. There is still considerable ambiguity aboutthe value of�B (between350 and700 MeV ) at which the critical point occurs inthese calculations.

2.2 Geometry and Space-Time Evolution of a Heavy-IonCollision

Nuclei which take part in a collision are objects of the finitevolume and hence theirgeometry has an important role in the understanding of the heavy-ion collisions. Both

2.2. GEOMETRY AND SPACE-TIME EVOLUTION OF A HEAVY-IONCOLLISION 7

Figure 2.2: The LQCD results for non-zero chemical potential [1] suggest theexistence of a critical point well aboveRHIC chemical potential values. Thesolid line represent a 1-st order phasetransition, while the dotted one indi-cates a crossover transition between twophases.

nuclei are Lorentz contracted in the direction of their relative movement (usually z-axisof the experiment) what leads to a high baryon density. The impact parameter vector~bconnects, in the transversex � y plane, the center of the target with the center of theprojectile and points from the target to the projectile. Itsmagnitude goes from 0 toRA +RB, whereRA andRB are the radii of the colliding nuclei. According to the magnitudeof the impact parameter vector one can distinguish peripheral (high magnitude of thevector~b) and central (small magnitude of the vector~b) collisions. In the case of peripheralcollisions, the overlapping region between the nuclei is minimal, while in the case of

b

b

participants

spectators

CENTRAL COLLISIONPERIPHERAL COLLISION

a) b)

Figure 2.3: A cartoon presentation of a central (a) and a peripheral (b) collision in x � z plane of thecollision.

central collisions it is maximal. In between these two extreme classes of the collisions areso called semicentral collisions. As an example, Fig. 2.3 shows as a cartoon a central (a)and a peripheral (b) collision inx� z plane of the collision.

Space-time evolution of a heavy-ion collision can be divided into three stages: Theearly stage of the collision, the stage of expansion and the freeze-out stage. An example


Time

Space

Freeze out

Mixedphase

QGP

Pre-equilibrium

Incoming nuclei

Interact inghadron gas

Figure 2.4: Schematic space-time evolution of a central heavy-ion collision with a QGP phase formedduring the collision.

with the formation of the QGP is shown in Fig. 2.4.

1. Early stage of the collision

Compression of matter in the early stage of the collision leads to increasing of en-ergy density. A part of the incident energy of the colliding nuclei is redistributedinto other degrees of freedom. A short time after the beginning of the collision fromthe highly excited QCD field appear secondary quarks. When a critical quark den-sity � is reached a transition from ’normal’ hadron matter into a color deconfinedmatter and restoring of chiral symmetry occurs. The energy density of the producedmedium is given by the Bjorken estimate [25]� = (dNdy )y=0 Eh�R2A�0 (2.2)where(dNdy )y=0 is the number of produced hadrons per unit rapidity at the midra-pidity, Eh is the average energy of produced hadrons,RA is the nuclear radius and�0 is the formation time of the medium which is not known but approximately it istaken to be 1 fm/c. In the case of central Pb+Pb collisions at

psNN = 17 GeV/c,the Bjorken estimate gives the initial energy density of 3.5GeV/fm3 [26] which is

2.3. HEAVY-ION COLLISIONS AND SIGNATURES OF THE QGP 9

much higher than the LQCD prediction of�1.0 GeV/fm3 where the phase transitionto deconfined quark and gluons occurs.

Given these large densities, in the created system of partons1under certain condi-tions multiple parton-parton collisions can establish a local thermodynamical equi-librium. If the time necessary for the thermalization is small enough in comparisonto the life of the system, the system can transit into a new phase, equilibrated QGP.The experiments with heavy-ion collisions should answer the following question:Are the interactions copious enough and rapid enough to thermalize the dynamicand expanding matter created in the laboratory?

2. Stage of expansion

Regardless of whether QGP was formed or not, the high pressure built up in thecollision will result in a fast expansion of the created system. If in the early stageof the collision the QGP was formed, during the expansion thetemperature of thesystem will decrease and at the critical temperatureT will appear a transition into amixed phase in which partons and hadrons exist together. In the mixed phase manypartonic degrees of freedom are redistributed into a smaller number of hadronicdegrees of freedom. In the hadronic phase, constituents of the system still interactand the system continues to expand. The temperature of the system in the hadronicphase is still very high (smaller thanT but higher than the freeze-out temperatureTfo).

3. Freeze-out stage

With the expansion the mean free path of particles becomes larger. The freeze-outappears when the mean free path becomes large enough. Then, by definition, thestrong interaction between the particles ceases and they continue to move freely.

Measuring observables as the phase-space distributions ofthe produced particles, theratio of multiplicities between different particles species, the anisotropic transverse flowand then-particle correlations (n = 2; 3; :::) one can get the information about differentevolution stages in nucleus-nucleus collisions.

2.3 Heavy-Ion Collisions and Signatures of the QGP

Although in the theoretical treatment of the thermodynamicand hydrodynamic behaviorof the QGP was done a lot, the complexity of heavy-ion collisions introduces significantquantitative ambiguities in deriving conclusions. Due to that one must identify the moststriking qualitative predictions of the QCD theory, which are able to survive the quantita-tive ambiguities and to look for a congruence of different observables which support suchpredictions.

In Fig. 2.5 is shown the phenomenological phase diagram of strongly interacting mat-ter [2, 27, 28] where different phases of the nuclear matter are present. In order to under-stand the nuclear EoS one has to measure parameters which govern the transition between

1Valence quarks, sea quarks and gluons have one common namepartons.


0.2 0.4 0.6 0.8 1 1.2 1.4

50

100

150

200

250

early universe

baryonic chemical potential µB [GeV]

tem

per

atu

re T

[MeV

]

RHIC

SPS

AGS

SIS

atomicnuclei neutron stars

chemical freeze-out

thermal freeze-out

hadron gas

quark-gluonplasma

deconfinementchiral restoration

Figure 2.5: The QCD phase diagram of the hadronic matter [2]. The points show chemical freeze-out ofhadrons extracted from different heavy-ion experiments.

different phases. The phase boundary can be constructing byequating chemical potential�B and pressurep between two phases. The properties of the system at the freeze-out arewell known from the systematic study of particle ratios. Theextracted freeze-out temper-atures and baryon chemical potentials from the experimental data with the incident energybeyond 10 AGeV are very close to the expected phase boundary.

2.4 Collective Flow as Signature of the QGP

Combining concepts from particle physics and nuclear physics gives a new approachin investigating the properties of matter and its interactions. In high energy physics(E=m� 1), interactionsare derived from gauge theories, and thematterconsists of par-

2.4. COLLECTIVE FLOW AS SIGNATURE OF THE QGP 11

tons2. In contrast, on nuclear physics scale, the strong interactions are shielded and canbe derived only in phenomenological theories, whereas the matter consists of extendedsystems which show collective behavior.

If initial interactions among the constituents are sufficiently strong to establish localthermal equilibrium and to maintain it during a significant evolution time, then the re-sulting matter may be treated as a relativistic fluid undergoing collective, hydrodynamicalflow. The hydrodynamical treatment for the description of fireballs formed in heavy-ioncollisions has a long history [29–32]. The details of the hydrodynamical evolution aresensitive to the EoS of the flowing matter. Hydrodynamics cannot be applied to matterwhich is not in a local thermal equilibrium, hence it must be supplemented with a phe-nomenological treatment for early and late stages of the collisions. The motion of anideal non-viscous fluid is completely described with the fluid velocity (~v), the pressure(p) and the energy and baryon density (� andnB). From the EoSp = p(�; nB) one canfind the slope�p�� which gives the square of the velocity sound (2s) exhibiting a high value(close to1=3) for the hadron gas and especially for the QGP, but has a soft point at themixed phase [31]. This softening of the EoS during the assumed phase transition hasconsequences to the system evolution.

In non-central collisions, the reaction zone has an almond shape which results in anazimuthally anisotropic pressure gradients. It produces anon-trivial elliptic flow pattern.Experimentally it is usually measured via a Fourier decomposition of the transverse mo-mentum distribution relative to the reaction plane which isdefined with the beam directionand the impact parameter vector~b. The important feature of the elliptic flow is the ”self-quenching” [33, 34] because the flowing of matter, induced bypressure, tends to reducespatial anisotropy and to increase momentum anisotropy. Due to that, the self-quenchingmakes the elliptic flow sensitive to early collision stages when the spatial anisotropy andpressure gradients are the biggest.

0 500 1000 1500 2000 2500 30000

0.02

0.04

0.06

0.08

0.1

LHC

v2 (EOS Q)

v2 (EOS H)

〈 〈 v⊥ 〉 〉 (EOS Q)

SPS RHIC

v2

total pion multiplicity density at y=0

Pb+Pb, b=7fm

0

0.2

0.4

0.6

0.8

1

〈 〈 v⊥〉 〉

Figure 2.6: Hydrodynamical pre-dictions of v2 excitation functions(left axis) and radial flow velocityhhv?ii (right axis) for non-centralPb+Pb collisions [3].

Calculations carried out for a fixed impact parameter vectoras a function of colli-sion energy (pT -integrated) show a dip starting at Super-Proton-Synchrotron (SPS) energy

2or hadrons if the energy is not large enough.


(Fig. 2.6 [3]). The existence of the dip reflects the softening of the EoS used. However,comparison of predicted (Fig. 2.6) with measured (Fig. 2.7 [4]) excitation functions forthe elliptic flow are subject to ambiguity concerning where and when the appropriate con-ditions of initial local thermal equilibrium for hydrodynamic applicability are actuallyachieved.

elliptic flow in Au+Au collisions

-0.1

0

0.1

10-1

1 10 102

103

104

Tlab (GeV/nucleon)

v 2 FOPIPlastic BallLANDEOSE895E877 pions, protonsNA49 pions, protonsCERES charged part.STAR charged part.

protons

in-plane

out-of-plane

Figure 2.7: The beam en-ergy dependence of the ellip-tic flow. The figure is takenfrom [4].

Another way to attain sensitivity of the elliptic flow to the EoS is the predicted ellipticflow magnitude on hadronpT and mass measured at a given collision energy and central-ity. The mass dependence has a simple kinematic origin, but the flow magnitude dependson EoS’s details. That is the reason why the pion, proton,� andK0S elliptic flow analysiswas performed in this thesis. Another reason for the� andK0S elliptic flow measurementis that the elliptic flow of strange particles can give an insight into very early stages of thecollisions.

Comparing the elliptic flow intensities between mesons and baryons one can have aninsight into mechanisms which govern the hadronization of the dense matter created inthe heavy-ion collisions. Certain scaling scenarios were developed in order to performsuch kind of investigation.

The energy-,pT - and mass-dependence of the elliptic flow is also affected byspecies-specific hadronic final state interactions close to the freeze-out where particles decouplefrom the system freely flying to the detector where hydrodynamics is not applicable any-more. A combination of macroscopic and microscopic models with hydrodynamics ap-plied at the early partonic and mixed-phase stages and hadronic transport models such asRQMD [35] at the later hadronic stage may offer a more realistic description of the wholeevolution than that achieved with a simplified sharp freeze-out treatment.

Chapter 3

METHODS IN FLOW ANALYSIS

In the investigations of the flow phenomena, different methods for the measurement of itsmagnitude were developed. First, quite shortly will be described two oldest methods, thesphericity tensor[36] and themean transverse momentumin the reaction plane [37]. Theconcept of thereaction planewill be presented gradually in Section 3.2. That concept isnecessary for the explanation of the two widely used methodsin the flow analysis: thealready mentioned mean transverse momentum and theFourier analysisof the particle az-imuthal distributions constructed with respect to the reaction plane [38–42]. The methodof Fourier analysis will be described as a generalization ofthe mean transverse momen-tum method. The presentation of the methods in flow analysis will be continued withthe method oftwo-particle correlations[43] which does not use the idea of the reactionplane for measuring the flow magnitude. In this thesis, for the Fourier analysis and for thetwo-particle correlations will be used a common name: the Standard Flow Analysis. Themethods of the Standard Flow Analysis are mainly insensitive to the non-flow effects. Inorder to distinguish flow from the non-flow contributions, J.-Y. Ollitrault developed thecumulantmethod [44, 45] and the method ofLee-Yang zeroes[46, 47]. These methodswill be presented in the last two sections of this Chapter.

3.1 Sphericity Tensor

As a first approach in the flow investigation appeared the method of the kinetic-energyflow tensor. This method is based on the construction of the second order spherical tensorF�� (which will be shortly named as sphericity tensor) defined inthe center-of-mass frameas F�� = MXi=1 pi�pi�=2mi; �; � = x; y; z (3.1)Here,pi� andmi are the momentum component and the mass of thei-th particle respec-tively. The sum goes over all particlesM detected in the collision. In that way,F��represents the total kinetic energy in the non-relativistic limit. The orientation and thevalues of the principal axes could be obtained by diagonalization of theF��. Thus, theevent shape in momentum space could be presented by such an ellipsoid. The spheric-ity tensor method also gives the information about the direction of the total momentum

13

14 CHAPTER 3. METHODS IN FLOW ANALYSIS

flow. The angle between the beam axis and the eigenvector associated with the largesteigenvalue of this tensor defines the flow angle�F which was used for quantifying themagnitude of the directed flow.

The sphericity tensor method was used for small colliding energies [36] while in thecase of high energies it is not used because the flow angle�F � hpxi=pbeam 1is too small(� 1) and the information about the differential flow magnitude is not provided.3.2 Mean Transverse Momentum in the Reaction Plane

The description of non-central collisions is more complicated than that of very centralcollisions because of presence of the azimuthal asymmetry in the initial state of the inter-action. Nevertheless, two colliding nuclei obey a reflection symmetry with respect to thereaction plane. Although the structure of the colliding nuclei gets destroyed, the reflectionsymmetry which was present in their initial state should be preserved during the collision.The momentum distribution of nucleons in the nuclei is isotropic in the transverse (x� y)laboratory plane. But the spatial distribution of matter does not have that property (forfinite impact parameter~b). Due to that, the momentum distribution evolves from isotropyinto an anisotropic shape but with an overall reflection symmetry. In Fig. 3.1 the spatial

planereaction

planereaction

space asymmetry

momentum asymmetry

∆x

∆yP=0

Pmax

Figure 3.1: A schematic view of a col-lision of two nuclei in the transverseplane. The spatial asymmetry, showed atthe top, is transformed into a momentumasymmetry, showed at the bottom, due tothe pressure gradient which was built upduring the collision.

asymmetry is represented by an overlapping zone (red area).The natural tendency of thematter is to flow in the direction with the highest pressure gradient. That as a consequenceproduce a collective anisotropical expansion of particlespreferentially emitted in the di-rection with the highest pressure gradient (Fig. 3.1 bottom). The transverse momentum

1hpxi will be defined in Section 3.2

3.2. MEAN TRANSVERSE MOMENTUM IN THE REACTION PLANE 15

method attempts to exploit and quantify possible anisotropies in the transverse momentaassociated with the reaction plane. Therefore, every eventshould be analyzed with respectto this initial azimuthal asymmetry. This asymmetry is fully determined by the reactionplane and the magnitude of the impact parameter vector~b. In the literature, the reactionplane is also known as the event plane.

The reaction plane is defined by the impact parameter vector~b and beam axis. If oneassumes that the beam axis corresponds to the longitudinal (z) axis of the experimentthen the reaction plane is determined by the angle� between the~b and thex axis of theexperiment (which could be arbitrarily chosen).

The transverse momentum method was proposed by Danielewiczand Odyniec [37].It was used over a wide range of colliding energies, from 25A MeV in the center-of-masssystem [48] to above 60A GeV [49]. In the original Danielewicz&Odyniec approach, theunderlying assumption is that particles are emitted in opposite direction in the forward(y > ym) and backward hemisphere (y < ym) owing to the momentum conservation.Then, the orientation of the reaction plane, i.e. the� angle, is determined by a vector~Qconstructed as ~Q = MXi=1 wi~pi;T ; (3.2)where the weightwi is 0 for pions and midrapidity protons,+1 for protons emitted inthe forward hemisphere (y � ym + Æ) and�1 for protons emitted in the backwardhemisphere2(y � ym � Æ). Such a choice of weights corresponded to the strong protondirected flow at low energies. The optimal value ofÆ could be determined by minimizationof the error in the determination of the reaction plane.

This method has been applied to evaluate directed flow in terms of the mean trans-verse momentum of particles projected into the reaction plane hpxi [37]. The obtaineddependence on the rapidity has a characteristicSshape and the slope of that curve givenby Fy = dhpxi=Ady (3.3)is used to quantify the strength of the directed flow. The normalization to the mass num-ber transforms the momentum into a velocity and this makes different particle speciescomparable. In addition to this flow signal, a quantitypdir� =PNi w � ~pT;i � ~Q=j ~Qj is usedto represent the total transverse momentum into the reaction plane.

3.2.1 The Reconstruction of the Reaction Plane

The orientation of the reaction plane is not knowna priori and as a first step in the flowanalysis it is necessary to reconstruct it. It can be reconstructed only if the outgoingparticles retains some memory of the initial collision geometry. So, the method uses theanisotropic flow itself to determine it. As there are different types of anisotropic flow, italso means that the reaction plane can be determined independently for each harmonic ofthe anisotropic flow. The reaction plane vector~Qn, which defines the reaction plane angle�n from then-th harmonic is defined by the equations

2The sign of the weight was arbitrarily chosen and it became a convention


Xn � Qn os(n�n) = MXi=1 wi os(n�i)Yn � Qn sin(n�n) = MXi=1 wi sin(n�i) (3.4)which give its components, or by�n = 1n artan� YnXn� (3.5)Here,�i represents the azimuthal angle of each outgoing particle. The sums go overall particlesi used for the reaction plane determination whilewi are their weights. Theweights are chosen in a way to make the reaction plane resolution as good as possible. Itcould be done by selecting the particles of one particular type, or weighting with rapidityin the case of the directed flow or with transverse momentum inthe case of the ellipticflow. Another way is to use the flow magnitude itself as a weight. It can be done in thefollowing way. In the first iteration one can use the weights mentioned above to performthe flow analysis. Then, in the second iteration, the flow analysis will be repeated usingthe obtained flow intensities as weights. It is obvious that for n = 1 andwi = pT Eq.(3.2) appears as a special case of the Eq. (3.4).

This is the general method for the reconstruction of the reaction plane orientation. Inpractice, several problems appear when one tries to use the reaction plane reconstructedwith such method to calculate the flow. The reason is the following. The flow calculationis always based on a correlation between a given particle with the reconstructed reactionplane. When a particle has been used in the calculation of thereaction plane, an auto-correlation effect appears. That effect is in the following. Whenever one measures theparticle azimuthal angle with respect to the reaction planereconstructed with Eq. (3.4),an autocorrelation of the given particle with itself appears if that particle is included insums of Eq. (3.4). In general, whenever one connects a particle with a construction inwhich that particle is used, the autocorrelation effect arises. The simplest way to removesuch an effect is to exclude such a particle from the above mentioned sums. That couldbe easily done if one saves the sums of sines and cosines from Eq. (3.4), and subtractsthe contribution of that particle from these sums. In case when one uses Eq. (3.2) for thereaction plane determination, the modified equation [50] isexpressed by~Qj = MX(j 6=)i=1wi~pT;i; (3.6)

In reality, e.g. in the case of electronic experiments, if one uses hits for the reactionplane reconstruction due to a possible hit splitting the above described method for re-moving the autocorrelation cannot help. One way to avoid theobstacle is to form tracksby matching hits from different detectors. The other way is to exchange the method. InSection 4.1 will be described so called slice method which isused in this analysis. Using

3.2. MEAN TRANSVERSE MOMENTUM IN THE REACTION PLANE 17

that method the autocorrelation effect, even in case of existence of artificial hit splitting,could be removed completely.

Another effect, the correlation due to the momentum conservation which appeared inthe directed flow analysis, could be removed already at the level of reconstruction of thereaction plane. At small energies, where this effect is rather strong, this could be doneby redefining of the Eq. (3.6) in the following way [50]. The system that determines thereaction plane~Qj is moving in the transverse direction with a momentum~pT = �~pT;j.Applying a boost~vb to each particlei,~vb = ~pT;imsys �mi (3.7)means that the system used to evaluate~Qj has no net transverse momentum. In Eq. (3.7)msys stands for the total mass of the system. Then Eq. (3.6) becomes~Q0j = MX(j 6=)i=1wi(~pT;i +mi~vb) (3.8)This procedure leads to a decreasing of the flow magnitude for1=M relative to the flowmagnitude obtained with Eq. (3.6) [51]. In Eq. (3.8)M is the number of particles used inthe reaction plane reconstruction.

3.2.2 Flattening of thedN=d� DistributionDue to the random distribution of the vector~b in the transversex � y plane, for an idealdetector the distribution of the azimuthal angle of the reaction planedN=d� has to beflat. In reality, different detector effects like the efficiency in� smaller than100%, or thegeometrical offset between the position of the beam and the center of the detector in thex�y plane make the distributiondN=d� not isotropic. Such an effect should be removedbefore doing flow analysis. There are several methods to remove such an effect.

The first one is to recenter the(Xn; Yn) distributions by subtracting(hXni; hYni) val-ues averaged over all events [52–54]. The main disadvantageof this method is that it isnot sensitive to anisotropies caused by higher harmonics. If such harmonics are presentone needs additional flattening of�n distributions. In the second method, one can con-struct the laboratory particle azimuthal distributions for all events and to use the inverseof this as weights in the Eq. (3.4) [52, 53, 55]. The limitation of this approach is that itdoes not take the multiplicity fluctuations around the mean value into account. A thirdmethod, the method of mixed events [52,53,55] is the next onewhich could be used. Theessence of the method is that one could divide the correlations of particles with respectto the non-flat (”raw”) reaction plane by the correlations ofparticles with respect to thereaction plane determined from another event. Such division can remove the correlationsdue to the acceptance. In the fourth method one can fit the non-flat distributions of thereaction plane angle�n to a Fourier expansion and to apply an event-by-event shifting ofthe reaction planes in order to make the final distributions isotropic [52,53]. The equation


for the shift is given by��n = 1n mmaxXm=1 2m [�hsin(mn�n)i os(mn�n) + hos(mn�n)i sin(mn�n)℄ (3.9)wheremmax is usually equal to 4.

In this analysis will be used the method of recentering and Fourier shifting subse-quently. The first method will make the raw reaction plane distribution roughly flat. Sub-sequent applying the Fourier shifting will make the reaction plane distribution completelyflat.

3.2.3 The Reaction Plane Resolution

Because the position of the true reaction plane is not knowna priori, one can only per-form Fourier decomposition of the invariant particle distributionE d3Nd3p with respect to thereconstructed position of the reaction plane�n wheren is the order of the harmonic fromwhich this position is reconstructed. Due to the finite multiplicity, the difference betweenthe true and the reconstructed reaction plane is not zero. So, the measured flow correla-tion has to be corrected for thatreaction plane resolution. The reaction plane resolutionis given by hos[n(�n � �)℄i (3.10)where� is the azimuthal angle of the true reaction plane. The resolution depends onthe flow harmonic and the flow itself. In order to calculate thevalue given by (3.10) oneconstructs reaction planes from two random subeventsa andb. The two random subeventsone gets by dividing the whole event into two pieces with verysimilar topology. In thiscase, the simple relationhos[n(�an � �bn)℄i = hos[n(�an � �)℄ihos[n(�bn � �)℄i (3.11)is valid. An important assumption here is that there are no other correlations except theones due to the flow. Eq. (3.11) allows to calculate the reaction plane resolution given by(3.10). For example, if one knows the correlations between two equal subevents then theresolution of each of them ishos[n(�an � �)℄i =phos[n(�an � �bn)℄i (3.12)If the two subevents are correlated, then the term inside thesquare-root in Eq. (3.12) ispositive. In Eq. (3.12) one calculates the reaction plane resolution of a subevent. Takinginto account that the multiplicity of the full event is twicelarger than the one of a subeventthen hos[n(�an � �)℄i =p2hos[n(�an � �bn)℄i (3.13)3.3 Two-particle Correlations

Wang [43] proposed to use two-particle azimuthal correlations in order to investigate theanisotropic flow. The idea is based on the fact that if (in the case of flow) particles are

3.4. FOURIER ANALYSIS OF THE AZIMUTHAL DISTRIBUTIONS 19

correlated to the reaction plane, then they are also mutually correlated. So, in presenceof non-zero flow, thePorr(��) distribution, constructed from two-particle correlationswith respect to the relative angle�� = �1 � �2 between two particles belonging tothe same event, is not flat. In reality, due to incomplete� acceptance in the detector ordue to finite efficiency for detecting particles at different�’s one has also to construct abackgroundPunorr(��) distribution in the same way asPorr(��), where now�� is thedifference between azimuthal angles of particles belonging to two different events. Onethen constructs the correlation function as a ratioC(��) � Porr(��)Punorr(��) (3.14)So, with this mixing technique the physical correlation between two particles is extractedwith elimination of the ’detector’ effects. In an ideal case, without non-flow effects, onehas C(��) = +1Xn=0 v2nein(��) = +1Xn=0 v2n os(n��) (3.15)a Fourier expansion of the measured correlation functionC(��) which gives the inte-grated flowvn.

For the differential flow,vdiffn , one has simply to replace�1 with the azimuthal angle of a particle in a narrow phase space window, andv2n in Eq. (3.15) is replaced withvnvdiffn .The crucial limitation of the two-particle correlation method is the impossibility to

separate the flow and non-flow correlations.

3.4 Fourier Analysis of the Azimuthal Distributions

The dependence on the particle emission azimuthal angle measured with respect to the(true) reaction plane angle (�) could be written as a Fourier expansion [38–42] of theinvariant particle distributionE d3Nd3pEd3Nd3p = 12� d2Nptdptdy 1 + 1Xn=1 2vn os[n(�� )℄! (3.16)Sine terms vanish due to the reflection symmetry with respectto the reaction plane. Themain advantage of the Fourier method, with respect to the sphericity method, is that themagnitude of the flow, which is characterized by the Fourier coefficientsvn, can be cor-rected for the reaction plane resolution, caused by the finite multiplicity of the event, bymultiplying the observed value ofvn with the inverse value of the reaction plane resolutiongiven by (3.13). This correction increases the value of the observed Fourier coefficients.Only the Fourier coefficients corrected for the reaction plane resolution can be comparedto the theoretical predictions, or to simulations filtered for the detector acceptance.

The Fourier coefficients in Eq. (3.16) are given byvn = hos[n(�� )℄i (3.17)


wherehi indicates an averaging over all particles of interest and over all events. A factor 2in front of eachvn in Eq. (3.16) is used in order to obtain a transparent physical meaningof the Fourier coefficients. In a coordinate system in which thex axis corresponds to aprojection of the reaction plane to the laboratory transverse plane are valid the followingformulae os(�� ) = px=pT ; sin(�� ) = py=pT (3.18)Then, according to Eq. (3.17), the coefficientv1 is equal tohpx=pT i and v2 is equalto h(px=pT )2 � (py=pT )2i. Now, it is obvious that the coefficientv1 corresponds to thedirected flow, andv2 to the elliptic flow.3.5 The Cumulants

The reaction plane� cannot be measured directly. As the correlation between each par-ticle and the reaction plane induces correlations among theparticles themselves,vn co-efficients could be experimentally measured from the azimuthal correlations between theoutgoing particles. These correlations are called ”flow correlations”. Both methods, thereaction plane and two-particle correlations are in use at intermediate and ultrarelativis-tic energies, but in both methods one usually assumes that the only source of azimuthalcorrelations is the flow. However, this assumption especially is not valid at SPS energies,where ”direct”, non-flow two-particle correlations become of the same magnitude as theflow correlations itself. Standard methods extract flow fromtwo-particle azimuthal cor-relations, either directly [43, 56], or through the correlation with respect to the reactionplane [37, 38, 41]. However, the correlation between two given particles is not only dueto the flow but also due to the other sources of correlation as quantum Bose-Einstein ef-fects, momentum conservation, resonance decays, jets, etc. When the flow is small, theseeffects may dominate the measured signal. The impact of the ’non-flow’ correlationson the flow analysis might be minimized by cuts in phase-spacewhich could be used toavoid the influence of quantum effects and resonance decays,while the contribution ofmomentum conservation can be calculated and subtracteda posteriori [57]. However,these various prescriptions require somea priori knowledge of non-flow correlations. So,it is necessary to assume thatall such sources of correlations are known and accountedfor, which may not be true. A new method of flow analysis, basedon a cumulant ex-pansion of multiparticle azimuthal correlations can overcome these difficulties [45]. Theprinciple of the method is that when the cumulants of higher order are considered, therelative contribution of non-flow effects, and thus the corresponding systematic errors,decreases. Denote by�j, wherej = 1; :::;M , the azimuthal angle of the particle detectedin an event with multiplicityM . Multiparticle azimuthal correlations could be generallywritten in the formhein(�1+:::+�k��k+1�::::��k+l)i, wheren is the Fourier harmonic understudy and the brackets indicate an average over all possiblecombinations ofk+ l particlesdetected in the same event and over all events. Correlationsbetweenk+ l particles couldbe decomposed into a sum of terms involving correlations between a smaller number ofparticles. For instance, two-particle correlationshein(�1��2)i can be written as:hein(�1��2)i = hein�1ihe�in�2i+ hhein(�1��2)ii (3.19)

3.5. THE CUMULANTS 21

wherehhein(�1��2)ii is by definition the second order cumulant. In order to understandthe physical meaning of the cumulant, consider a ’perfect’ detector, i.e. a detector withan isotropic acceptance. Then, the averagehein�ji vanishes due to the symmetry since�j is measured in the laboratory, not with respect to the reaction plane. The first term inthe right-hand side (r.-h.s.) of Eq. (3.19) vanishes and thecumulant reduces to the mea-sured two-particle correlations. The importance of cumulants appears at a more realisticcase of a non-perfect detector. Then the first term on the r.-h.s. of the Eq. (3.19) can benon-vanishing. But the cumulant vanishes if�1 and�2 are uncorrelated. Then the cumu-lanthhein(�1��2)ii isolates the physical correlation and disentangles it fromtrivial detectoreffects. There are several physical contributions to the correlationshhein(�1��2)ii whichseparate into flow and non-flow correlations. When the sourceis isotropic (there is noflow), only direct correlation remains. Direct correlationscales with the multiplicityMlike 1=M [57,58]. So, the correlation between two arbitrary pions isproportional to1=M .If there is a flow, a correlation between emitted particles and the reaction plane, it gener-ates azimuthal correlations between any two outgoing particles, and gives a contributionv2n to the second order cumulant. One can measure the flow using second order cumulantif this contribution dominates over the non-flow contribution, i.e. ifvn � 1=pM [57,58].This is the domain of validity of the standard flow analysis, which is based on two-particlecorrelations.

3.5.1 Integrated Flow

The main benefit of the use of cumulants is in construction of higher order cumulants andseparation flow from non-flow correlations. To illustrate itconsider a perfect detector anddecompose the measured four-particle correlation as:hein(�1+�2��3��4)i = hein(�1��3)ihein(�2��4)i+ hein(�1��4)ihein(�2��3)i+hhein(�1+�2��3��4)ii (3.20)where two first terms in the r.-h.s. comes from possible two-particle combinations. Theremaining termhhein(�1+�2��3��4)ii is the fourth-order cumulant by definition. Althoughit is insensitive to two-particle non-flow correlations it could be sensitive to higher ordernon-flow correlations, i.e. direct four-particle correlations. Fortunately their probabilityis very small. Due to the symmetry between�1 and�2 (�3 and�4) the Eq. (3.20) can berewritten as: hein(�1+�2��3��4)i = 2hein(�1��3)i2 + hhein(�1+�2��3��4)ii (3.21)So, in principle it is possible to construct an expression for the4th order cumulant whicheliminates both detector effects and non-flow correlations.

Generating Functions

Even without assuming a perfect detector, cumulants could be expressed via generatingfunctions. The generating functionGn(z) is a real valued function of a complex variable

22 CHAPTER 3. METHODS IN FLOW ANALYSISz = x + iy defined as:Gn(z) = MYj=1[1 + wjM (z?ein�j + ze�in�j )℄ (3.22)in each event, wherez? = x � iy is the complex conjugate ofz andwj is a statisticalweight chosen in some way. This function could be averaged over events with the samemultiplicity M . The expansion of such a function into power series generates correlationsto all orders: hGn(z)i = ::: = 1 + zhe�in�1i+ z?hein�1i+ M � 1M ��z22 he�in(�1+�2)i+ z?22 hein(�1+�2)i+ zz?hein(�1��2)i�+ ::: (3.23)In this way theaveragedgenerating functionhGn(z)i contains all the information aboutmultiparticle azimuthal correlations.

In the case of the perfect detector,hGn(z)i does not depend on the phase ofz, butonly on its magnitudejzj = px2 + y2. If one changesz into zein�, the only effect isa shift of all angles by the same quantity. But, as the probability that one event occursunder a global rotation is unchanged, one concludes thathGn(z)i is invariant under sucha transformation.

Thegeneratingfunction of the cumulants is defined as:Cn(z) �M�hGn(z)i1=M � 1� (3.24)and its expansion into power series ofz andz? defines the cumulantsCn(z) �Xk;l z?kzlk!l! hhein(�1+:::+�k��k+1�:::��k+l)ii (3.25)If particles are uncorrelated, all the cumulants beyond order one are vanished. Indeed, ifall �j in Eq. (3.22) are independent from each other, then the mean value of the productis the product of the mean valueshGn(z)i =M�1 + 1M (z?hein�i+ zhe�in�i))�M (3.26)Then the generating function of the cumulants reduces toCn(z) = z?hein�i+ zhe�in�i) (3.27)Comparing it with Eq. (3.25) all cumulants of order higher than1 vanishes, as it is ex-pected in the case of uncorrelated particles.

The interesting cumulants are the diagonal terms withk = l which are related to theflow. They will be denoted withnf2kg:nf2kg � hhein(�1+:::+�k��k+1�:::��2k)ii (3.28)


In practice, it is rather difficult to expand the generating functionCn(z) analytically be-yond the order2. The simplest way to extractnf2kg is to computeGn(z) and thenCn(z)for various values ofz in order to tabulate it. Then one has to interpolate their successivederivatives from the obtained matrix. For example, one way is to use the points:zp;q = xp;q + iyp;q; (3.29)xp;q � r0pp os( 2q�qmax ); (3.30)yp;q � r0pp sin( 2q�qmax ) (3.31)for p = 1; :::; kmax andq = 1; :::; qmax � 1 whereqmax > 2kmax. Typical values arekmax = 3 andqmax = 7. As one wants to know the behavior ofGn(z) andCn(z) near theorigin, r0 has to be a small number. AveragingCn(zp;q) over phase ofz one obtainsCp � 1qmax qmax�1Xq=0 Cn(zp;q); p = 1; :::; kmax (3.32)It is shown [45] thatCp is related to the cumulantsnf2kg through the linear systemCp = kiXk=1 (r0pp)2k(k!)2 nf2kg (3.33)which can be resolved in order to extract the cumulants. Withki = 3 it givesnf2g = 1r20 (3C1 � 32C2 + 13C3) (3.34)nf4g = 2r40 (�5C1 + 4C2 � C3) (3.35)nf6g = 6r60 (3C1 � 3C2 + C3) (3.36)The relations between the cumulantsnf2kg and the integrated flowVn, or to be moreprecise their estimatesVnf2kg are given byVnf2g2 = nf2g; (3.37)Vnf4g4 = �nf4g; (3.38)Vnf6g6 = nf6g4 (3.39)Statistical Errors

Due to the finite number of eventsN , the reconstructed integrated flow has a statisticalfluctuation which could be calculated from the covariance matriceshVnf2kgVnf2lgi �hVnf2kgihVnf2lgi. The covariance matrices contain information on the standard error.


In [45] it is shown that in the case of a huge flow, i.e.,Vn � 1=pM , the followingequation is valid hVnf2kgVnf2lgi � hVnf2kgihVnf2lgi = 12MN (3.40)In this limit, reconstructed flow from different cumulant orders coincide and the error is1=p2MN independently ofk. This is easy to understand: when flow is large comparedto 1=pM the reaction plane can be reconstructed with a high accuracy.

In the general case, whenVn and 1=pM are of the same order of magnitude thestatistical deviations are given by (ÆVnf2g)2 = 12MN 1 + 2�22�2(ÆVnf4g)2 = 12MN 1 + 4�2 + �4 + 2�62�6 (3.41)(ÆVnf6g)2 = 12MN 3 + 18�2 + 9�4 + 28�6 + 12�8 + 24�1024�10where(ÆVnf2kg)2 = hVnf2kg2i�hVnf2kgi2 and�2 �MV 2n . In the limit of a very largeflow (�� 1) all three equations reduce to Eq. (3.40).

In the case of a very weak flow, i.e.,Vn � 1=pM , different estimates ofVn areuncorrelated, and hence flow cannot be reconstructed and statistical errors loose theirsense.

3.5.2 Differential Flow

When the integrated (over phase space) flow valuesVnf2kg are known, one can measurethe ”differential flow”, i.e. flow value in a narrower phase space window. Following thenotation in [45], let’s call a particle belonging to the given narrower window a ’proton’(although it can be anything else). It’s azimuthal angle is denoted with , and it’s dif-ferential flow asvn(pT ; y) = hein( ��)i. The particles used for the integrated flowVnare ’pions’. Once the integrated flowVn is known, one can reconstruct the differentialflow vn from the correlations between the azimuth and ’pion’ azimuths�j. In order todo that, first one constructs a generating function between the ’proton’ and ’pions’. Thisfunction is the average value over all ’protons’ ofeip Gn(z), whereGn(z) is defined withEq. (3.22) evaluated for the event to whom the ’proton’ belongs. Note that the averageprocedure is not exactly the same as in the case of the integrated flow. One must firstaverage over ’protons’ in the same event (i.e. with the sameGn(z)) and then to averageover only those events where there are ’protons’.

Expanding in power series ofz andz?, one obtains:heip Gn(z)i = heip i+ zhei(p �n�1)i+ z?hei(p +n�1)i+ ::: (3.42)which generates azimuthal correlations between the ’proton’ and arbitrary number of ’pi-ons’. The generating function is thenDp=n(z) � heip Gn(z)ihGn(z)i (3.43)


Note that the ’proton’ should not be one of the ’pions’ in order to avoid of autocorrela-tions, and while the number of ’pions’ in Eq. (3.22) is fixed, the number of ’protons’ isallowed to fluctuate from event to event.

The cumulants are, by definition, the coefficients in the power series of the generatingfunction, i.e. Dp=n(z) �Xk;l z?kzlk!l! hheip +in(�1+:::+�k��k+1�:::��k+l)ii (3.44)The physical meaning of these cumulants is the same as in the case of the integrated flow.They eliminate the detector effects and the lower order non-flow correlations, so only thedirect (non-flow) correlations of orderM�k�l remain.

If the ’proton’ is not correlated with the ’pions’, then Eq. (3.43) becomeDp=n(z) =heip i for any z and all cumulants are vanishing. In the case when the correlation ispresent, expanding Eq. (3.43) up to orderz and comparing to Eq. (3.44) one obtainshhei(p �n�1)ii � hei(p �n�1)i � heip ihein�1i (3.45)what is analogous to Eq. (3.19) and has the same interpretation, namely that the cumulantmethod gives exactly the same result as the two-particle correlations.

In the case of a perfect detector all cumulants defined in Eq. (3.44) are real, be-cause reversing the sign of all azimuthal angles ! � ; �j ! ��j leavingDp=n(z)unchanged. Also the transformation�j ! ��j changesz into z? in Gn(z), soDp=n(z) � he�ip Gn(z?)ihGn(z?)i (3.46)Comparing it with Eq. (3.43) one sees thatz has been changed intoz? and into � .SinceGn(z) is a real function one finally obtainsDp=n(z) = D?p=n(z?). From that oneconcludes that the coefficients in Eq. (3.44) are real. In thecase of a real detector, theyare complex, but only the real part has a physical meaning. Writing p = mn the relevantquantities are:dmn=nf2k +m + 1g � Re[hhein(m +�1+:::+�k��k+1�:::��2k+m)ii℄ (3.47)whereRe denotes the real part, andf2k + m + 1g denotes correlations between one’proton’ and2k+m ’pions’. The cumulantdmn=nf2k+m+1g has a contribution from flowproportional tovpV 2k+mn . In that way one can calculate the differential flowvp from thecumulantdmn=nf2k+m+1g knowing a previously calculated value of the integrated flowVn. In order to avoid the trivial autocorrelation effect, the ’proton’ must not be one of the’pions’. The same problem is well known in the Standard Flow Analysis where the wayto exclude it was to reject the particle under the study (in this case the ’proton’) from thedefinition of the sums (Eq. (3.4)) which were used for the reaction plane reconstruction.In the method of cumulants one simply removes the ’proton”s contribution by dividingGn(z) with 1+(z?ein + ze�in )=M in the numerator of the Eq. (3.43). As in the case of


the integrated flow, a practical way to determine the differential flow consists in tabulatingthe generating functionDp=n(z) at the ’pions’zp;q given by Eq. (3.29)Dp � (r0pp)mqmax qmax�1Xq=0 [os(m 2q�qmax )Xp;q + sin(m 2q�qmax )Yp;q℄ (3.48)with Xp;q + iYp;q � Dp=n(zp;q) = 1N 0 Pev:w:prot:[Pprot: os(p )Gn(z)p;q℄1Nevts PevtsGn(zp;q) ++i 1N 0 Pev:w:prot:[Pprot: sin(p )Gn(z)p;q℄1Nevts PevtsGn(zp;q) (3.49)whereN 0 is the total number of ’protons’. There is a relation betweenthe cumulantdmn=nf2k +m+ 1g and numbersDp via the system:Dp = kd�1Xk=0 (r0pp)2k+mk!(k +m)! dp=nf2k +m+ 1g; 1 � p � kd (3.50)which can be solved in cumulantsdp=nf2k+m+1g. For instance, withkd = 2 andm = 1which is used forv1=1 andv2=2, one has:dn=nf2g = 1r20 (2D1 � 12D2) (3.51)dn=nf4g = 1r40 (�2D1 +D2) (3.52)while for kd = 2 andm = 2 is used to calculatev2=1,d2n=nf3g = 1r40 (4D1 � 12D2) (3.53)d2n=nf5g = 1r60 (�6D1 + 32D2) (3.54)When the cumulants are determined in that way, then they mustbe related to the differen-tial flow vmn=n. In the case of a perfect detector following equations are validvn=nf2g = dn=nf2gVn (3.55)vn=nf4g = �dn=nf4gV 3n (3.56)v2n=nf3g = d2n=nf3gV 2n (3.57)v2n=nf5g = �d2n=nf5g2V 4n (3.58)

3.6. THE LEE-YANG ZEROES 27

Statistical Errors

Although, as in the case of the integrated flow, the reconstructed differential flow dependson the number of eventsNevts (denominator in Eq. (3.49)), this depends additionally onthe number of ’protons’N 0 (numerator in Eq. (3.49)) in a narrow phase-space windowwhere it is measured. Hence, one can neglect the contribution from the denominator tothe statistical error of the differential flow.

Again, in the case of the weak flow, i.e.,vn � 1=pM , correlations between theestimations from the different orders vanish and the statistical errors loose their sense.Whenvn � 1=pM (large flow), covariance matrix reduces tohvmn=nf2k +m+ 1gvmn=nf2l +m+ 1gi �hvmn=nf2k +m + 1gihvmn=nf2l +m+ 1gi = 12N 0 (3.59)

In the general case, whenvn and 1=pM are of the same order of magnitude, thefollowing equations are valid. Form = 1:hvn=nf2g2i � hvn=nf2gi2 = 12N 0 1 + �2�2 (3.60)hvn=nf2gvn=nf4gi � hvn=nf2gihvn=nf4gi = 12N 0 (3.61)hvn=nf4g2i � hvn=nf4gi2 = 12N 0 2 + 6�2 + �4 + �6�6 (3.62)where�2 �Mv2n. Form = 2:hv2n=nf3g2i � hv2n=nf3gi2 = 12N 0 2 + 4�2 + �4�4 (3.63)hv2n=nf3gv2n=nf5gi � hv2n=nf3gihv2n=nf5gi = 12N 0 3 + �2�2 (3.64)hv2n=nf5g2i � hv2n=nf5gi2 = 12N 0 6 + 24�2 + 9�4 + 10�6 + 4�84�8 (3.65)

In the limit of very large flow (�� 1) all six equations reduce to Eq. (3.59).3.6 The Lee-Yang Zeroes

The Lee-Yang zeroes method [46, 47] derived by J.-Y. Ollitrault, N. Borghini and R.S.Bhalerao is based on the genuine correlation between a largenumber of particles. It ismore natural and more reliable than all other methods which have been used so far. Sincethe anisotropic flow appears as a collective effect, involving all particles produced in anevent, it is indeed natural to characterize it by means of a global multiparticle observable.All previously used methods were practically based on2k-particle correlations (where2k � M ) and so they are not the appropriate tool to probe a collective behavior. Espe-cially, in all these methods, except the cumulant method, non-flow effects were neglected.


3.6.1 Integrated Flow

In order to measure genuine anisotropic flow one first defines the integrated flow asVn = h MXj=1 wj os[n(�j � �)℄i (3.66)where the sum goes over all particles detected in an event andwj are appropriate weights.In the Eq. (3.66)� is the azimuth of the impact parameter vector~b. The integratedflow is connected with an average of the Fourier coefficientvn via: Vn = Mwvn, whereMw =PMj=1 wj.

In order to compute the integrated flow one has to compute for each event the complex-valued function: g�(ir) = MYj=1[1 + irwj os(n(�j � �)℄ (3.67)for various values of the real positive variabler and of the angle� (0 � � � �=n

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